Auxiliary functions for molecular integrals with Slater‐type orbitals. I. Translation methods

Many types of molecular integrals involving Slater functions can be expressed, with the ζ‐function method in terms of sets of one‐dimensional auxiliary integrals whose integrands contain two‐range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the ζ‐function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two‐center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals. II. Two‐center expansion method

Using expansion formulas for the charge‐density over Slater‐type orbitals (STOs) obtained by the one of authors [I. I. Guseinov, J Mol Struct (Theochem) 1997, 417, 117] the multicenter molecular integrals with an arbitrary multielectron operator are expressed in terms of the overlap integrals with the same screening parameters of STOs and the basic multielectron two‐center Coulomb or hybrid integrals with the same operator. In the special case of two‐electron electron‐repulsion operator appearing in the Hartree–Fock–Roothaan (HFR) equations for molecules the new auxiliary functions are introduced by means of which basic two‐center Coulomb and hybrid integrals are expressed. Using recurrence relations for auxiliary functions the multicenter electron‐repulsion integrals are calculated for extremely large quantum numbers. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 117–125, 2001     

Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals

Multicenter integrals over noninteger n Slater type orbitals with integer and noninteger values of indices u of screened Coulomb type potentials, f(u)(eta,r)=r(u-1)e(-etar), and their first and second derivatives with respect to Cartesian coordinates of the nuclei of a molecule are described. Using complete orthonormal sets of Psi(alpha) exponential type orbitals and rotation transformation of two-center overlap integrals, these integrals are expressed through the noncentral potential functions depending on the molecular auxiliary functions A(k) and B(k). The series expansion formulas derived for molecular integrals of screened Coulomb potentials and their derivatives are especially useful for the computation of multicenter electronic attraction, electric field, and electric field gradient integrals. The convergence of series is tested for arbitrary values of parameters of potentials and orbitals.     

Evaluation of expansion coefficients for translation of Slater‐type orbitals using complete orthonormal sets of exponential‐type functions

By the use of exponential‐type functions (ETFs) the simpler formulas for the expansion of Slater‐type orbitals (STOs) in terms of STOs at a displaced center are derived. The expansion coefficients for translation of STOs are presented by a linear combination of overlap integrals. The final results are of a simple structure and are, therefore, especially useful for machine computations of arbitrary multielectron multicenter molecular integrals over STOs that arise in the Hartree–Fock–Roothaan approximation and also in the Hylleraas correlated wave function method for the determination of arbitrary multielectron properties of atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 126–129, 2001     

Analytical Evaluation of Multicenter Multielectron Integrals of Central and Noncentral Interaction Potentials over Slater Orbitals Using Overlap Integrals and Auxiliary Functions

Using expansion formulas for central and noncentral interaction potentials (CIPs and NCIPs, respectively) in terms of Slater type orbitals (STOs) obtained by the author (I.I. Guseinov, J. Mol. Model., in press), the multicenter multielectron integrals of arbitrary interaction potentials (AIPs) are expressed through the products of overlap integrals with the same screening parameters and new auxiliary functions. For auxiliary functions, the analytic and recurrence relations are derived. The relationships obtained for multicenter multielectron integrals of AIDs are valid for the arbitrary quantum numbers, screening parameters and location of orbitals.     

Calculation of molecular electric and magnetic multipole moment integrals of integer and noninteger n Slater orbitals using overlap integrals

Closed formulas are established for the magnetic multipole moment integrals of integer and noninteger n Slater‐type orbitals (ISTOs and NISTOs) in terms of electric multipole moment integrals for which the analytic expressions through the overlap integrals with ISTOs and NISTOs are derived. The overlap integrals are evaluated by the use of auxiliary functions. Using the derived expressions the multipole moment integrals, and therefore the electric and magnetic properties of molecules, can be evaluated most efficiently and accurately. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater‐type orbitals with noninteger principal quantum numbers

The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Calculation of electric multipole moment integrals with the different screening parameters via the Fourier transform method

Three‐center electric multipole moment integrals over Slater‐type orbitals (STOs) can be evaluated by translating the orbitals on one center to the other and reducing the system to an expansion of two‐center integrals. These are then evaluated using Fourier transforms. The resulting expression depends on the overlap integrals that can be evaluated with the greatest ease. They involve expressions for STO with different screening parameters that are known analytically. This work gives the overall expressions analytically in a compact form, based on Gegenbauer polynomials. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

Evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals with integer and noninteger principal quantum numbers

A general formula has been established for the expansion of the product of two normalized associated Legendre functions centered on the nuclei a and b. This formula has been utilized for the evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals (STOs) with integer and noninteger principal quantum numbers. The formulas given in this study for the evaluation of two‐center overlap and nuclear attraction integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, orbital exponents, and internuclear distances. © 2001 Wiley Periodicals, Inc. Int J Quantum Chem, 2001     

Improved partition–expansion of two‐center distributions involving slater functions

The calculation of the electronic structure of large systems is facilitated by the substitution of the two‐center distributions by their projections on auxiliary basis sets of one‐center functions. An alternative is the partition–expansion method in which one first decides what part of the distribution is assigned to each center, and next expands each part in spherical harmonics times radial factors. The method is exact, requires neither auxiliary basis sets nor projections, and can be applied to Gaussian and Slater basis sets. Two improvements in the partition–expansion method for Slater functions are reported: general expressions valid for arbitrary quantum numbers are derived and the efficiency of the procedure is increased giving analytical solutions to integrals previously computed by numerical quadrature. The efficiency of the new version is assessed in several molecules and the advantages over the projection methods are pointed out. © 2013 Wiley Periodicals, Inc.     

Symbolic Calculation of Two‐Center Overlap Integrals Over Slater‐Type Orbitals

Two‐center overlap integrals over Slater type orbitals (STOs) have been expressed in terms of the well‐known Mulliken's integrals B_n(pt) using Rodrigues's formula for normalized associated Legendre functions. A computer program is written in Mathematica 4.0 for the evaluation of two‐center overlap integrals over STOs. Using this computer program, symbolic tables are presented for two‐center overlap integrals up to quantum numbers 1 ≤ n,n′ ≤ 3, 0 ≤ l,l′ ≤ 2, ?2 ≤ m,m′ ≤ 2. Numerical results of this work, for some quantum sets, have also been compared with prior literature and best agreement achieved with recent works of Barnett while some discrepancies were obtained with works of Öztekin et al. and Guseinov et al.     

Evaluation of one‐electron molecular integrals over complete orthonormal sets of Ψα ‐ETO using auxiliary functions

By the use of expansion and one‐range addition theorems, the one‐electron molecular integrals over complete orthonormal sets of Ψ^α ‐exponential type orbitals arising in Hartree–Fock–Roothaan equations for molecules are evaluated. These integrals are expressed through the auxiliary functions in ellipsoidal coordinates. The comparison is made using Slater‐, Coulomb‐Sturmian‐, and Lambda‐type basis functions. Computation results are in good agreement with those obtained in the literature. The relationships obtained are valid for the arbitrary quantum numbers, screening constants, and location of orbitals. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Computation of two‐center Coulomb integrals over Slater‐type orbitals using elliptical coordinates

A general analytic formula is obtained for the two‐center Coulomb integrals over Slater‐type orbitals in elliptical coordinates. Finite series expansions are used in the evaluation of the radial part of the integrals. The analytic formula is expressed in terms of a product of the well‐known auxiliary functions A_k(p) and B_k(p) and incomplete gamma functions. Recursive relations for the computer evaluation of these functions are given as well. The recursive relations are stable and our computer results are in good agreement with the benchmark values given in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Contracted auxiliary Gaussian basis integral and derivative evaluation

The rapid evaluation of two-center Coulomb and overlap integrals between contracted auxiliary solid harmonic Gaussian functions is examined. Integral expressions are derived from the application of Hobson's theorem and Dunlap's product and differentiation rules of the spherical tensor gradient operator. It is shown that inclusion of the primitive normalization constants greatly simplifies the calculation of contracted functions corresponding to a Gaussian multipole expansion of a diffuse charge density. Derivative expressions are presented and it is shown that chain rules are avoided by expressing the derivatives as a linear combination of auxiliary integrals involving no more than five terms. Calculation of integrals and derivatives requires the contraction of a single vector corresponding to the monopolar result and its scalar derivatives. Implementation of the method is discussed and comparison is made with a Cartesian Gaussian-based method. The current method is superior for the evaluation of both integrals and derivatives using either primitive or contracted functions.     

On the calculation of arbitrary multielectron molecular integrals over Slater‐type orbitals using recurrence relations for overlap integrals I. Single‐center expansion method

The multicenter charge‐density expansion coefficients [I. I. Guseinov, J Mol Struct (Theochem) 417 , 117 (1997)] appearing in the molecular integrals with an arbitrary multielectron operator were calculated for extremely large quantum numbers of Slater‐type orbitals (STOs). As an example, using computer programs written for these coefficients, with the help of single‐center expansion method, some of two‐electron two‐center Coulomb and four‐center exchange electron repulsion integrals of Hartree–Fock–Roothaan (HFR) equations for molecules were also calculated. Accuracy of the results is quite high for the quantum numbers, screening constants, and location of STOs. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 146–152, 2000     

Calculation of multicenter electronic attraction,electric field and electric field gradient integrals of Coulomb potential over integer and noninteger <Emphasis Type="Italic">n</Emphasis> Slater orbitals

With the help of complete orthonormal sets of - ETOs, where = 1,0, – 1, – 2, ... a large number of series expansion formulas for the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of integer and noninteger n Slater type orbitals (ISTOs and NISTOs) is established through the overlap integrals with the same screening constants and the new central and noncentral interaction potentials depending on the coordinates of the nuclei of a molecule are introduced. The convergence of the series is tested by calculating concrete cases for arbitrary quantum numbers, screening constants and location of ISTOs and NISTOs.     

Explicit and implicit multi‐center integrations

We present truncated expansions of multicenter one‐electron nuclear attraction and two‐electron repulsion integrals over localized basis functions in terms of one‐ and two‐center integrals of “Coulomb,” “exchange,” and “hybrid” type. Two variants are discussed: the “Explicit Multi‐center Integrations” and the “Implicit Multi‐Center Integrations” (abbreviated as “EMCI” and “IMCI”, respectively). While EMCI also deals with individual integrals, the IMCI option is the more appealing one: it enables us to evaluate the entire matrix elements of “Restricted Hartree–Fock”‐type in a very effective and chemically meaningful way. Due to the diatomic nature of our expansions, integrations over “Slater‐Type Orbitals” become well‐feasible, too. © 2012 Wiley Periodicals, Inc.     

Evaluation of Multicenter Electronic Attraction, Electric Field and Electric Field Gradient Integrals with Screened and Nonscreened Coulomb Potentials over Integer and Noninteger n Slater Orbitals

By the use of complete orthonormal sets of -exponential-type orbitals, where ( = 1, 0, –1, –2,...) the multicenter electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals of nonscreened and Yukawa-like screened Coulomb potentials are expressed through the two-center overlap integrals with the same screening constants and the auxiliary functions introduced in our previous paper (I.I. Guseinov, J. Phys. B, 3 (1970) 1399). The recurrence relations for auxiliary functions are useful for the calculation of multicenter EA, EF and EFG integrals for arbitrary integer and noninteger values of principal quantum numbers, screening constants, and location of slater-type orbitals. The convergence of the series is tested by calculating concrete cases.